Determinants

Determinants determine whether a square matrix is invertible. This is a huge help to us, as we will see. In the literature, you will see either a function abbreviation for the determinant or vertical bars, like so:

The determinant is a function from ℂ^{n × n} to ℂ. In other words, it takes an n × n square matrix as input and spits out a scalar. For a 1 × 1 matrix, the determinant is just the number (easy enough). For a 2 × 2 matrix, this is the formula. You should probably just commit it to memory if you can:

I will give you exercises at the end of this section to help with the memorization part, which will also give you a feel for the determinant itself.

There is a method for calculating determinants for bigger matrices, but it is rather involved, and once you've mastered 2 × 2 matrices, I would suggest using a matrix calculator. It's just like arithmetic; you should...